Topic 3.7 Inference and Experiments: use the presence or absence of random selection and random assignment to determine the scope of inference, that is, whether results generalize to a population and whether a causal conclusion is justified.

What this topic is asking

The College Board (Topic 3.7) wants you to use the presence or absence of two randomisations, random selection and random assignment, to determine the scope of inference: whether a result can be generalized to a population, and whether a causal conclusion is justified.

The two questions, the two randomisations

These are genuinely separate. A study can have one, both, or neither. Confusing them is the most common error in the topic, so it is worth fixing the pairing firmly: selection to generalize, assignment to cause. Selection is about who is in the study; assignment is about what is done to them.

The four-quadrant grid

Most real experiments use volunteers (random assignment, no random selection), so they support causation for those volunteers but not automatic generalization. Most surveys use random selection but are observational (no random assignment), so they support generalisable associations but not causation. Recognizing which quadrant a study falls in, then stating exactly the conclusion that quadrant allows, is the core skill Topic 3.7 assesses.

Why each randomisation does its job

Random selection works because a chance-chosen sample is, on average, a fair miniature of the population, so a statistic computed from it estimates the population parameter with only quantifiable random error; that is what makes extrapolation to the population legitimate. Random assignment works because distributing subjects to treatments by chance tends to make the groups alike in every variable except the treatment, so confounding is eliminated by design and a difference in response can be pinned on the treatment. The two mechanisms are independent: making the sample representative says nothing about whether treatments were imposed fairly, and balancing treatment groups says nothing about whether the subjects resemble a wider population. That independence is exactly why the conclusions they license are independent, and why you must check for each one separately before writing a conclusion.

Writing a defensible conclusion

On the exam, a conclusion about scope should do three things: state whether the result generalizes (and to whom), state whether a causal claim is justified, and tie each judgement to the presence or absence of the relevant randomisation, in context. For a volunteer experiment with random assignment, write that the treatment caused the observed difference for these subjects, but that without random selection the finding may not extend to a broader population. For a random-sample survey, write that the association can be generalized to the sampled population, but that because treatments were not assigned, a confounding variable could be responsible, so no causal claim is warranted. Disciplining yourself to name the randomisation that does (or does not) support each half of the conclusion is what separates a full-credit answer from a vague one, and it is the reasoning every later inference unit assumes you can produce.

Try this

Q1. A study randomly selects subjects but does not randomly assign treatments. What kind of conclusion is justified? [2 points]

• Cue. A generalisable association (random selection allows generalization), but no causation, because without random assignment a confounding variable could explain the link.

Q2. Why can a volunteer experiment with random assignment still not generalize to a population? [1 point]

• Cue. It lacks random selection, so the volunteers may not represent any wider population; random assignment gives causation for the subjects but not generalization.